For guaranteeing the fast convergence of row and column methods, one crucial point is to introduce a practical and appropriate criterion used to select the working rows or columns from the coefficient matrix. In this work we present two relaxed greedy deterministic row and column methods. Our approaches are based on several ideas and tools, including Petrov-Galerkin conditions and the promising adaptive index selection method, i.e., relaxed greedy selection strategy, which further generalize the RGRK [1]}, RGRCD [2]} and FDBK [3]} methods. Our convergence analyses reveal that the resulting algorithms all have the linear convergence rates, which are bounded by the explicit expressions.
